5. Exchange interaction The dipole interaction between the magnetic moments of the electrons is much too weak for supporting magnetic order of materials at high temper- atures. For explaining the observed magnetism, we need to find a strong interaction between the electrons. We might think that an interaction de- pending explicitly on the spin (the magnetic moments) of the electrons is needed; no such interaction is known though. In 1928, Werner Heisenberg realized that the responsible interaction is the Coulomb repulsion between electrons; this is a strong interaction but does not explicitly depend on the spin. Spin selectivity is due to quantum mechanics, in particular the Pauli principle: Two electrons with parallel or antiparallel spin behave dif- ferently even though the fundamental interaction is the same; the spatial * * wave function ψ(r1, r2) has to be antisymmetric and symmetric, respec- tively. One consequence is that two electrons with parallel spin cannot be in the same place. In order to discuss how the Coulomb interaction term leads to an exchange interaction of the spins, we first write this interaction in second quantization; we will introduce this very useful representation first. 5.1 Occupation number representation for fermions So far, we have discussed the Hamiltonian in so-called first quantization: * Ne Ne 2 Ne Pi * * * H = H0 + H1 , H0 = hi = + V(ri) , H1 = u(ri − rj) 2m Xi=1 Xi=1 Xi=1 Xi<j (5.1) * * where u(ri − rj) is either the bare Coulomb potential 2 * * e u(ri − rj) = * * (5.2) |ri − rj| or an effective, screened interaction. First quantization implies that the an- tisymmetry of the wave function has to be taken into account by working 44 with Slater determinants which is rather cumbersome. Therefore, many- body calculations in solid state theory are usually performed in so-called second quantization, using the occupation number representation. Slater determinant Let’s assume that we can solve the single particle problem exactly, i.e. for electron i we have h |k (i) = ε |k (i) (5.3) i αi kα αi or in position representation *2 * p * * * h ϕ (r ) = i + V(r ) ϕ (r ) = ε ϕ (r ) (5.4) i kα i 2m i kα i kα kα i * with a complete set of single particle quantum numbers kα = (l, k, σ) of Bloch states. The Pauli principle now implies that only the part of the product space of Ne single particle Hilbert spaces is realized which consists of the particle indices of totally antisymmetric wave functions. A basis of the Ne particle Hilbert space are the Slater determinants which we can compose of single particle states as follows: 1 χP (1) (Ne) |ψk , ,k (1 Ne) = (−1) |kP(1) |kP(N ) 1 ··· Ne ··· i √N ! i ··· e i e P S X∈ Ne (1) (N ) |k1 |k1 e 1 .i ··· i. 1 (i) = det . = det |kα (5.5) √N ! √N ! i e |k (1) |k (Ne) e Ne Ne i ··· i where P are the elements of the permutation group SNe of Ne elements, and χP is the character of the permutation (number of transpositions, which lead to the permutation). The product state |k (1)|k (2) |k (Ne) 1i 2i ··· Ne i means that particle 1 is in state k1, particle 2 in state k2 and so on; but as the particles are not distinguishable, it has to be irrelevant which particle is in state k1, k2 etc. Therefore, we have to sum over all possible permutations. The Slater determinants are a suitable basis for the Ne particle Hilbert space HA(Ne), even if not all states of this Hilbert space correspond to a single Slater determinant. 45 Fock space The basis of HA(Ne) which is described by Slater determinants can also be written down in occupation number representation by writing down how many of the indistinguishable Ne particles are in state kα; however, the sum over all occupation numbers has to be Ne. We can get rid of this restriction if we do not work in the Ne particle Hilbert space but instead in Fock space H = H (0) H (1) H (N ) (5.6) A,Fock A ⊕ A ⊕ · · · ⊕ A e ⊕ · · · which is defined as direct sum over the Hilbert spaces for all possible par- ticle numbers. If we allow an arbitrary number of (identical) particles in the Hilbert space, then this product space is called Fock space. We can now define operators that “ascend” and “descend” between segments of Fock space with different particle numbers. These operators create and annihilate particles; therefore, they are called creation and annihilation operators. They play a central role in many serious calculations within quantum mechanics. Fock space is always explicitly or implicitly used for grand canonical treatments. In the following, we note the most important relations for fermions and bosons; therefore, we use N for the number of particles. Starting point is the representation of N particle states. Let’s assume a discrete, ordered single particle basis is given: |1 , |2 , . , where i in |i i *i i stands for a set of single particle quantum numbers (lkσ)i. The normaliza- tion is i|j = δ . All N particle states can be represented by superposition h i ij of P |r1 |r2 ... |rN (5.7) ± i i i where P+ symmetrizes for bosons and P− antisymmetrizes for fermions. Explicitly, we have 1 P |r |r ... |r = (−1)P|r |r ... |r (5.8) − 1i 2i Ni √ P(1)i P(2)i P(N)i N! P S X∈ N and 1 P+ |r1 |r2 ... |rN = |rP(1) |rP(2) ... |rP(N) (5.9) i i i √N!n !n !... i i i 1 2 P S X∈ N where P runs over all permutations and ni is the number of single particle states |i in the product. i 46 An equivalent characterization of the (basis) states is possible in occupation number representation: {n} |n1, n2,... := P |1 ... |1 |2 ... |2 ... (5.10) ≡ i ± i i i i n1 times n2 times (Obviously, for fermions we have |n {z{0,} 1|}). {z{n}} thus stands for a com- i ∈ plete set of occupation numbers for all single particle states. A further step for the efficient representation is the introduction of particle creation and annihilation operators: ci†, ci. We do this for fermions here and cite the result for bosons below. We define ci and ci† by their effect on the basis states as follows: i−1 nj ci {n} = ci| ... ni ... = (−1) j=1 | ... (ni − 1) ... i Pi−1 n i † { } †| j=1 j | ci n = ci ... ni ... = (−1) ... (ni + 1) ... (5.11) i P i Concerning the notation, ci† is indeed the adjunct operator for ci: nj (−1) j<i , if mi = ni − 1 , mj = nj for j = i , {m} ci {n} = P 6 0, otherwise mj (−1) j<i , if ni = mi + 1 , nj = mj for j = i , {n} ci† {m} = P (5.12)6 0, otherwise Remark: c i maps totally antisymmetric N particle states to totally anti- symmetric (N − 1) particle states; ci† acts “in the opposite way”: Creation and annihilation, respectively. For the particle operators the important anticommutation relations are valid: [ci, cj]+ = cicj + cjci = 0 (5.13) and also [ci†, cj†]+ = ci†cj† + cj†ci† = 0 (5.14) Verification: we assume i < j (otherwise, the labels can be exchanged, i = j is obvious): νj cicj {n} = ci(−1) | ... (nj − 1) ... νi+νj i = (−1) | ... (ni − 1) ... (nj − 1) ... ν i c c {n} = c (−1) i | ... (n − 1) ... j i j i i 47 = (−1)νj−1+νi | ... (n − 1) ... (n − 1) ... (5.15) i j i i−1 Here we abbreviated νi = j=1 nj. Thus, the second equation of (5.15) has an extra minus sign, and if we add both equations, we find cicj + cjci = 0. Furthermore, P [ci, cj†]+ = δij (5.16) Justification: We assume i < j (otherwise, we again relabel); as before, we find νi+νj cicj† {n} = (−1) | ... (ni − 1) ... (nj + 1) ... ν −1+ν i † { } j i | cj ci n = (−1) ... (ni − 1) ... (nj + 1) ... (5.17) i i.e. for i = j c and c† anticommute. Now let i = j: 6 i j {n} , if ni = 0 , cici† {n} = 0, if ni = 1 , 0, if ni = 0 , ci†ci {n} = (5.18) {n} , if ni = 1 , From the sum of these two equations, we obtain (cic† + c†ci)|{n} = {n} y [ci, c†]+ = 1 . i i i i If we now define the vacuum as |0 := |00 . 0 . (5.19) i i then we have n n {n} = (c†) 1(c†) 2 ... |0 , (5.20) 1 2 i so that for N particle states, we obtain c† c† ... c† |0 , (5.21) r1 r2 rN i Therefore, we have the correspondence, for example for two particles 1 |k1 |k1 ψk1k2(12) = i i c1†c2†|0 (5.22) √2 |k2 |k2 ↔ i i i † †| † †| ψk1k2(12) = − ψk1k2(12) c2c1 0 = −c1c2 0 (5.23) ↔ i i 48 Both for the Slater determinant in first quantization and in second quan- tization, the antisymmetry of the wave function is guaranteed. In particular for fermions, this has the consequence c†c†|0 = −c†c†|0 = 0 , 1 1 i 1 1 i two fermions cannot have the same quantum numbers. For bosons, we briefly note the definitions c | ... n ... = √n | ... (n − 1) ... i i i i i i c†| ... n ... = √n + 1| ... (n + 1) ... (5.24) i i i i i i as well as the commutation relations [ci, cj] = [ci†, cj†] = 0 [ci, cj†] = δij (5.25) Particle number operator From the anticommutation (commutation) relations, we can also conclude that the operator ni = ci†ci (for bosons: ni = bi†bi) is the particle number operator.